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Theorem cfinufil 22825
Description: An ultrafilter is free iff it contains the Fréchet filter cfinfil 22790 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
cfinufil (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ Fin} ⊆ 𝐹))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑋

Proof of Theorem cfinufil
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elpwi 4522 . . . . 5 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
2 ufilb 22803 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
32adantr 484 . . . . . . . . 9 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → (¬ 𝑥𝐹 ↔ (𝑋𝑥) ∈ 𝐹))
4 ufilfil 22801 . . . . . . . . . . . 12 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
54adantr 484 . . . . . . . . . . 11 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝐹 ∈ (Fil‘𝑋))
6 filfinnfr 22774 . . . . . . . . . . . . 13 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑋𝑥) ∈ 𝐹 ∧ (𝑋𝑥) ∈ Fin) → 𝐹 ≠ ∅)
763exp 1121 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → ((𝑋𝑥) ∈ 𝐹 → ((𝑋𝑥) ∈ Fin → 𝐹 ≠ ∅)))
87com23 86 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘𝑋) → ((𝑋𝑥) ∈ Fin → ((𝑋𝑥) ∈ 𝐹 𝐹 ≠ ∅)))
95, 8syl 17 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ Fin → ((𝑋𝑥) ∈ 𝐹 𝐹 ≠ ∅)))
109imp 410 . . . . . . . . 9 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → ((𝑋𝑥) ∈ 𝐹 𝐹 ≠ ∅))
113, 10sylbid 243 . . . . . . . 8 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → (¬ 𝑥𝐹 𝐹 ≠ ∅))
1211necon4bd 2960 . . . . . . 7 (((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) ∧ (𝑋𝑥) ∈ Fin) → ( 𝐹 = ∅ → 𝑥𝐹))
1312ex 416 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ Fin → ( 𝐹 = ∅ → 𝑥𝐹)))
1413com23 86 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ( 𝐹 = ∅ → ((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
151, 14sylan2 596 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → ( 𝐹 = ∅ → ((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
1615ralrimdva 3110 . . 3 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ → ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
174adantr 484 . . . . . . . . . . . 12 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → 𝐹 ∈ (Fil‘𝑋))
18 uffixsn 22822 . . . . . . . . . . . 12 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → {𝑦} ∈ 𝐹)
19 filelss 22749 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ {𝑦} ∈ 𝐹) → {𝑦} ⊆ 𝑋)
2017, 18, 19syl2anc 587 . . . . . . . . . . 11 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → {𝑦} ⊆ 𝑋)
21 dfss4 4173 . . . . . . . . . . 11 ({𝑦} ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
2220, 21sylib 221 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) = {𝑦})
23 snfi 8721 . . . . . . . . . 10 {𝑦} ∈ Fin
2422, 23eqeltrdi 2846 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin)
25 difss 4046 . . . . . . . . . . 11 (𝑋 ∖ {𝑦}) ⊆ 𝑋
26 filtop 22752 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
27 elpw2g 5237 . . . . . . . . . . . 12 (𝑋𝐹 → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
2817, 26, 273syl 18 . . . . . . . . . . 11 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ {𝑦}) ⊆ 𝑋))
2925, 28mpbiri 261 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋)
30 difeq2 4031 . . . . . . . . . . . . 13 (𝑥 = (𝑋 ∖ {𝑦}) → (𝑋𝑥) = (𝑋 ∖ (𝑋 ∖ {𝑦})))
3130eleq1d 2822 . . . . . . . . . . . 12 (𝑥 = (𝑋 ∖ {𝑦}) → ((𝑋𝑥) ∈ Fin ↔ (𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin))
32 eleq1 2825 . . . . . . . . . . . 12 (𝑥 = (𝑋 ∖ {𝑦}) → (𝑥𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
3331, 32imbi12d 348 . . . . . . . . . . 11 (𝑥 = (𝑋 ∖ {𝑦}) → (((𝑋𝑥) ∈ Fin → 𝑥𝐹) ↔ ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
3433rspcv 3532 . . . . . . . . . 10 ((𝑋 ∖ {𝑦}) ∈ 𝒫 𝑋 → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
3529, 34syl 17 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → ((𝑋 ∖ (𝑋 ∖ {𝑦})) ∈ Fin → (𝑋 ∖ {𝑦}) ∈ 𝐹)))
3624, 35mpid 44 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → (𝑋 ∖ {𝑦}) ∈ 𝐹))
37 ufilb 22803 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑦} ⊆ 𝑋) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
3820, 37syldan 594 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (¬ {𝑦} ∈ 𝐹 ↔ (𝑋 ∖ {𝑦}) ∈ 𝐹))
3918pm2.24d 154 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (¬ {𝑦} ∈ 𝐹 → ¬ 𝑦 𝐹))
4038, 39sylbird 263 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → ((𝑋 ∖ {𝑦}) ∈ 𝐹 → ¬ 𝑦 𝐹))
4136, 40syld 47 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑦 𝐹) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → ¬ 𝑦 𝐹))
4241impancom 455 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)) → (𝑦 𝐹 → ¬ 𝑦 𝐹))
4342pm2.01d 193 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)) → ¬ 𝑦 𝐹)
4443eq0rdv 4319 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)) → 𝐹 = ∅)
4544ex 416 . . 3 (𝐹 ∈ (UFil‘𝑋) → (∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹) → 𝐹 = ∅))
4616, 45impbid 215 . 2 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹)))
47 rabss 3985 . 2 ({𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ Fin} ⊆ 𝐹 ↔ ∀𝑥 ∈ 𝒫 𝑋((𝑋𝑥) ∈ Fin → 𝑥𝐹))
4846, 47bitr4di 292 1 (𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ Fin} ⊆ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wne 2940  wral 3061  {crab 3065  cdif 3863  wss 3866  c0 4237  𝒫 cpw 4513  {csn 4541   cint 4859  cfv 6380  Fincfn 8626  Filcfil 22742  UFilcufil 22796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1o 8202  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fbas 20360  df-fg 20361  df-fil 22743  df-ufil 22798
This theorem is referenced by: (None)
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